The Crystal Field Model

The hybrid orbitals required for tetrahedral, square planar, and linear complex ions. The metal ion hybrid orbitals are empty, so the metal ion bonds to the ligands by accepting lone pairs. 

A set of six hybrid orbitals on can accept an electron pair from each of six NHj ligands to form the Co(NHj)6 ion. 

A linear complex requires two hybrid orbitals 180° from each other. This arrangement is given by an sp hybrid set (Fig. 19.20). Thus in the linear Ag(NH3)2 ion the Ag is described as sp hybridized. 

Although the localized electron model can account in a general way for metal-ligand bonds, it is rarely used today because it cannot predict important properties of complex ions, such as magnetism and color. Thus we will not pursue the model any further. 

The main reason that the localized electron model cannot fully account for the properties of complex ions is that in its simplest form it gives no information about how the energies of the d orbitals are affected by complex ion formation. This is critical because, as we will see, the color and magnetism of complex ions result from changes in the energies of the metal ion d orbitals caused by the metal-ligand interactions. 

Group theory provides the symmetry basis for MO theory, and the symmetry labels Cg and t g have a particular significance which those who are familiar with chemical applications of group theory will appreciate. Those unfamiliar with group threory should simply accept these symbols as labels used to denote the symmetries of particular sets of orbitals. In the symmetry of an octahedron the directions. v, r and z are equivalent and cannot be distinguished one from another. In terms of symmetry,. v. v and r must be treated as a set of three inseparable entities. Similarly, in an octahedral environment, the d, d and d orbitals cannot be distinguished 

This ionic model provides a good basis for explaining anomalies in the variation of lattice energies and hydration energies across the first-row d-block for octahedral metal ions. As electrons are added to the t 

Q Using the 0D(/ values below estimated from spectroscopic measurements. calculate the crystal held stabilization energy of [Fe(NH,)J in kJ mol ( 0D(/ = 20.000 cm assume a pairing energy of 19.000 cm and that 1 k.I mol = 83 cm ). 

A This involves comparing 0Dq with the pairing energy (PE) to determine that the complex is low spin. The crystal field splitting diagram shows the electron configuration from which the value of the CFSE can be calculated  

Remember to take into account the number of udditionul pairing energies required in low-spin cases before converting units from cm to k.F mol to izive CFSE = - 24.1 k.F mol 

We are now finally ready to tackle the question of why copper sulfate is blue. We begin by writing the electron configuration of copper  

Because copper is present in copper sulfate as the plus two ion, we next remove the two most energetic electrons from the copper atom. These electrons are the 4s electrons (the 4s-orbital has a higher energy than the 3d-orbitals in ions). (When the electron configuration is written in order of increasing n value, we always remove the electrons from the outermost orbital as written in order to form the ion.) So, the electron configuration of the copper plus two ion is  

Because the d-electrons are the outermost electrons in the ion they are most likely to be responsible for bonding or color. In general, the outermost electrons are known as the valence electrons. We must now focus our attention on them. 

The electron configuration written above is the appropriate description of the electrons in the isolated ion for example, an ion in the gaseous phase where it is not close to any other ion. But in the solid, the d-orbitals will have somewhat different energies. Let us assume 

Provide a diagram of the surroundings of a copper ion in solid copper sulfate. 

To understand the effect of this difference, we need to consider which type of orbital is lower in energy. Because the negative point-charge ligands repel negatively charged electrons, the electrons will prefer the d orbitals farthest from the ligands. In other words, the dxz, dyz, and dxy orbitals (called 

An octahedral arrangement of point-charge ligands and the orientation of the 3d orbitals. 

The energies of the 3d orbitals for a metal ion in an octahedral complex. The 3d orbitals are degenerate (all have the same energy) in the free metal ion. In the octahedral complex the orbitals are split into two sets as shown. The difference in energy between the two sets is designated as A (delta). 

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We will discuss the crystal field model here. It assumes that the bonding between metal "Crystal field" isn t a very descriptive term,... 

Most coordination compounds are brilliantly colored, a property that can be explained by the crystal field model. 

Two symmetry parameterizations of the angular overlap model of the ligand field. Relation to the crystal field model. C. E. Schaffer, Struct. Bonding (Berlin), 1973,14, 69-110 (33). 

We are concerned with what happens to the (spectral) d electrons of a transition-metal ion surrounded by a group of ligands which, in the crystal-field model, may be represented by point negative charges. The results depend upon the number and spatial arrangements of these charges. For the moment, and because of the very common occurrence of octahedral coordination, we focus exclusively upon an octahedral array of point charges. 

Schaffer CE (1968) A Perturbation Representation of Weak Covalent Bonding. 5 68-95 Schaffer CE (1973) Two Symmetry Parameterizations of the Angular-Overlap Model of the Ligand-Field. Relation to the Crystal-Field Model. 14 69-110 Scheldt WR, Lee YJ (1987) Recent Advances in the Stereochemistry of Metallotetrapyrroles. 64 1-70... 

The second row of Table 6 gives the net populations of the 3 and Ap atomic orbitals. They are quite different from those predicted by the crystal field model, given in the first row. 

Schaffer, C.E. Two Symmetry Parameterizations of the Angular-Overlap Model of the Ligand-Field. Relation to the Crystal-Field Model. Vol. 14, pp. 69-110. 

The general influence of covalency can be qualitatively explained in a very basic MO scheme. For example, we may consider the p-oxo Fe(III) dimers that are encountered in inorganic complexes and nonheme iron proteins, such as ribonucleotide reductase. In spite of a half-filled crystal-field model), the ferric high-spin ions show quadrupole splittings as large as 2.45 mm s < 0, 5 = 0.53 mm s 4.2-77 K) [61, 62]. This is explained... 

A quantitative consideration on the origin of the EFG should be based on reliable results from molecular orbital or DPT calculations, as pointed out in detail in Chap. 5. For a qualitative discussion, however, it will suffice to use the easy-to-handle one-electron approximation of the crystal field model. In this framework, it is easy to realize that in nickel(II) complexes of Oh and symmetry and in tetragonally distorted octahedral nickel(II) complexes, no valence electron contribution to the EFG should be expected (cf. Fig. 7.7 and Table 4.2). A temperature-dependent valence electron contribution is to be expected in distorted tetrahedral nickel(n) complexes for tetragonal distortion, e.g., Fzz = (4/7)e(r )3 for com-... 

Both Fe(ll)(TPP) and Fe(II)(OEP) have positive electric quadrupole splitting without significant temperature dependence which, however, cannot be satisfactorily explained within the crystal field model [117]. Spin-restricted and spin-unrestricted Xoi multiple scattering calculations revealed large asymmetry in the population of the valence orbitals and appreciable 4p contributions to the EFG [153] which then was further specified by ab initio and DFT calculations [154,155]. 

Modelling the Magnetic Properties of Lanthanide Single-Ion Magnets The Use of the Crystal Field Model... 

The crystal field model may also provide a calciflation scheme for the transition probabilities between levels perturbed by the crystal field. It is so called weak crystal field approximation. In this case the crystal field has little effect on the total Hamiltonian and it is regarded as a perturbation of the energy levels of the free ion. Judd and Ofelt, who showed that the odd terms in the crystal field expansion might connect the 4/ configuration with the 5d and 5g configurations, made such calculations. The result of the calculation for the oscillator strength, due to a forced electric dipole transition between the two states makes it possible to calculate the intensities of the lines due to forced electric dipole transitions. 

Some individual compounds have been studied using LCAO-MO theory in the Wolfsberg-Helmholz approximation (5). Although this method is somewhat more realistic and allows one to account for other properties (such as "charge-transfer bands, EPR, and NMR experiments) nevertheless, compared to the crystal field model it is much more laborious, it is only vahd for the individual case, and the choice of parameters in often rather arbitrary. 

The application of the angular overlap method to MXg chromophores of trigonal bipyramidal and square p3u-amidal stereochemistry leads to the patterns of Fig. 2 for the energies of the antibonding "d molecular orbitals (dc). The crystal field model leads to a similar pattern. 

Van Vleck also pointed out that even in the case of very highly covalent bonding [as in, e.g., Ni(CO)4 or Fe(CN)J ], which is best treated by using MO theory, the symmetry properties and requirements remain exactly the same as for the crystal field model and the ligand field model. 

Although the physical basis of the crystal field model is seen to be unsound, the fact remains that, in summarizing the importance of the symmetry of the ligand environment, it qualitatively reproduces many of the features of the magnetic and spectral properties of transition metal complexes. This early qualitative success established its nomenclature in the fields of these properties. While we shall have little more to say about crystal field theory as such, much of the rest of this article will be couched in the language of the crystal field model, and for that reason some little trouble has been taken to outline its development. 

Entirely general analytical expressions for the matrix elements of equation (4) have been listed for the d-orbital case for an almost arbitrary assembly of charges surrounding a metal atom.5,38 They are reproduced in Appendix 1. By implementing these expressions as a computer program the problem of calculating the d-orbital energies in the crystal field model for any ordinary stereochemistry is made trivial. 

The sequence of energy levels obtained from a simple molecular orbital analysis of an octahedral complex is presented in Fig. 1-12. The central portion of this diagram, with the t2g and e levels, closely resembles that derived from the crystal field model, although some differences are now apparent. The t2g level is now seen to be non-bonding, whilst the antibonding nature of the e levels (with respect to the metal-ligand interaction) is stressed. If the calculations can be performed to a sufficiently high level that the numerical results can be believed, they provide a complete description of the molecule. Such a description does not possess the benefit of the simplicity of the valence bond model. 

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